A Random Differential Equation Approach for Modeling the Growth of Microalgae in Photobioreactors

The three-state photosynthetic factory model is frequently employed to analyze microalgal growth in photobioreactors, wherein cells continuously transition between light and dark areas. Experimental data indicate that hydrodynamic mixing results in non-constant, randomly varying intervals between successive light-dark transitions. To address this characteristic, we reformulate the deterministic model as a random differential equation, regarding the switching period as a positive random variable. We derive closed-form expressions for the long-term mean and variance of the model, showing that the average random model differs from the quasi-steady periodic deterministic trajectory. Monte Carlo simulations are utilized to illustrate how the distribution of switching periods and the time fraction spent in darkness influence productivity. The simulations show that, if the average irradiance per cycle matches the optimum under continuous illumination, rapid flashing maintains high productivity even under highly variable periods, while slower and irregular cycles can lead to significant losses, with adjustments to dark fractions having a relatively minor effect. To the best of our knowledge, this work provides the first systematic application of a random differential equation framework to a PSF-type model for microalgal growth under intermittent light regimes, and offers quantitative guidance for the design and operation of photobioreactors under realistic, highly variable light conditions.